Fast linear response algorithm for differentiating chaos

Abstract

We devise a new algorithm, called the linear response algorithm, for differentiating SRB measures with respect to system parameters, where SRB measures are fractal limiting stationary measures of chaotic systems. The algorithm is illustrated on an example which is difficult for previous algorithms.The algorithm works for chaos on general manifolds with any unstable dimension, u. The algorithm is efficient and robust: its main cost is solving u many first-order and second-order tangent equations, and it does not compute oblique projections. The convergence to the true derivative is proved for uniform hyperbolic systems.The core of our algorithm is the first numerical treatment of the unstable divergence, a central object in the linear response theory for fractal attractors. We give a new characterization that the unstable divergence can be expressed by the solution of a renormalized second-order tangent equation, whose second derivative is taken in a modified shadowing direction, computed by the non-intrusive shadowing algorithm.

Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

“…The second map, in Eq. 40, is parameterized by a single real-valued parameter s. It was constructed in [32] by adding one additional expanding rotation and extra interaction terms between contracting and expanding directions of the Smale-Williams map used in modeling of oscillating circuits [35]. If s is moderately low, this map has two positive LEs (m = 2), with values close to log 2 and log 3, and a negative one.…”

“…In case of the solenoid map, we notice a significant peak right after the beginning of the recursion. This is a consequence of the randomly chosen initial condition x 0 that is likely to be located beyond the attractor, given its complex geometry [32].…”

Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems

“…The S3 algorithm is one method to efficiently compute Ruelle's formula (2). Ni's Linear Response Algorithm [Ni20] is another alternative that converges to (2), wherein the computations of the non-intrusive least squares shadowing algorithm [NW17] and a particular characterization of the unstable divergence are leveraged to provide a fast computation of (2). Ni's Linear Response Algorithm proposes to split Ruelle's formula into shadowing and an unstable contribution while we propose a different decomposition in the S3 algorithm (see [CW21, Appendix A], [Ni20] for a more detailed comparison of these two algorithms).…”

“…Ni's Linear Response Algorithm [Ni20] is another alternative that converges to (2), wherein the computations of the non-intrusive least squares shadowing algorithm [NW17] and a particular characterization of the unstable divergence are leveraged to provide a fast computation of (2). Ni's Linear Response Algorithm proposes to split Ruelle's formula into shadowing and an unstable contribution while we propose a different decomposition in the S3 algorithm (see [CW21, Appendix A], [Ni20] for a more detailed comparison of these two algorithms). Abramov and Majda [AM07] developed a blended response technique for (2) whereby an ensemble sensitivity estimate [LHAH02, EHL04] -a direct numerical approximation of (2) -is used for responses to stable components of the perturbations.…”

“…But despite their importance, the lack of an efficient and rigorous computation of Ruelle's linear response formula [Rue97,Rue03] has hindered downstream applications of linear response in high-dimensional chaotic models encountered in practice. Recently, some new methods have been proposed to evaluate decomposed and regularized versions of Ruelle's formula [CW21,Ni21,CW20]. The purpose of this paper is the rigorous proof of one such method, the space-split sensitivity (S3) algorithm [CW21].…”